Calderón-Zygmund operators on compact Lie groups
Let \(G\) be a connected, simply connected, compact semisimple Lie group of dimension \(n\), let \(g\) be the Lie algebra of \(G\), and let \(d\) be a bi-invariant metric on \(G\) induced by the Killing form on the complexification of \(g\). Suppose \(0 < \alpha < n\) and \(r > 0\). A convolution kernel of type \((\alpha, r)\) is a function \(K\) on \(G\) which is of class \(C^r\) on \(G \backslash \{e\}\) and satisfies \[ \biggl |X^J \bigl( K(x) \bigr) \biggr |\leq C_J d(x,e)^{\alpha - n - |J |} \tag{1} \] for all \(J = (j_1, \dots, j_n)\) with \(|J |\leq r\) and \(x \neq e\), where \(X^J = X_1^{j_1} X_2^{j_2} \dots X_n^{j_n}\). \(X_1, \dots, X_n\) is an orthogonal basis of \(g,e\) is the identity of \(G\), and \(C_J\) is a constant independent of \(x \in G\). Define the Calderón-Zygmund operator \(T_K\) with kernel \(K\) by \[ (T_Kf) (x) = \int_G f(y) K(xy^{-1}) dy, \quad x \in G, \] for \(f \in L^1 (G)\). If \(r\) is a positive integer, then a kernel of type \((0,r)\) is a distribution \(K\) on \(G\) which is of class \(C^r\) on \(G \backslash \{e\}\), satisfies (1) with \(\alpha = 0\), and guarantees the boundedness of \(T_K\) on \(L^2 (G)\). -- The main result of the paper is devoted to the boundedness of \(T_K\) on the real Hardy spaces \(H^p (G)\), \(p > 0\). Theorem: Suppose \(r\) is a positive integer and \(0 \leq \alpha < n\). If \(K\) is a kernel of type \((\alpha, r)\) and \({\alpha \over n} < {1 \over p_1} = {1 \over p_2} + {\alpha \over n} < 1 + {r \over n}\), then \(T_K\) extends to a bounded operator from \(H^{p_1} (G)\) to \(H^{p_2} (G)\).
scientific article; zbMATH DE number 704926